图书信息

作者简介

内容简介

目录

图书信息

出版社: 世界图书出版公司; 第2版 (2009年4月1日)

外文书名: Modular Functions and Dirichlet Series in Number Theory (2nd Edition)

平装: 204页

正文语种: 英语

开本: 24

ISBN: 7510004403, 9787510004407

条形码: 9787510004407

尺寸: 22 x 14.8 x 1.2 cm

重量: 299 g

作者简介

作者：(美国) 阿波斯托尔 (Apostol.T.M.)

内容简介

《数论中的模函数和狄利克莱级数(第2版)》讲述了：This is the second volume of a 2-volume textbook* which evolved from a course (Mathematics 160) offered at the California Institute of Technology during the last 25 years.The second volume presupposes a background in number theory com-parable to that provided in the first volume, together with a knowledge of the basic concepts of complex analysis

目录

Chapter1 Ellipticfunctions

1.1 Introduction

1.2 Doublyperiodicfunctions

1.3 Fundamentalpairsofperiods

1.4 Ellipticfunctions

1.5 Constructionofellipticfunctions

1.6 TheWeierstrassfunction

1.7 TheLaurentexpansionofganeartheorigin

1.8 Differentialequationsatisfiedbyξ

1.9 TheEisensteinseriesandtheinvariantsg2andg3

1.10 Thenumberse1,e2,e3

1.11 ThediscriminantA

1.12 Klein'smodularfunctionJ(τ)

1.13 InvarianceofJunderunimodulartransformations

1.14 TheFourierexpansionsofg2(τ)andg3(τ)

1.15 TheFourierexpansionsof△(τ)andJ(τ)

ExercisesforChapter1

Chapter2 TheModulargroupandmodularfunctions

2.1 M6biustransformations

2.2 Themodulargroup

2.3 Fundamentalregions

2.4 Modularfunctions

2.5 Specialvaluesof

2.6 Modularfunctionsasrationalfunctionsof

2.7 Mappingpropertiesof

2.8 ApplicationtotheinversionproblemforEisensteinseries

2.9 ApplicationtoPicard'stheorem

ExercisesforChapter2

Chapter3 TheDedekindetafunction

3.1 Introduction

3.2 Siegei'sproofofTheorem3.1

3.3 Infiniteproductrepresentationfor△(τ)

3.4 Thegeneralfunctionalequationforη(τ)

3.5 Iseki'stransformationformula

3.6 DeductionofDedekind'sfunctionalequationfromIseki'sformula

3.7 PropertiesofDedekindsums

3.8 ThereciprocitylawforDedekindsums

3.9 CongruencepropertiesofDedekindsums

3.1 0TheEisensteinseriesG2(τ)

ExercisesforChapter3

Chapter4 Congruencesforthecoefficientsofthemodularfunctionj

4.1 Introduction

4.2 ThesubgroupFo(q)

4.3 FundamentalregionofFo(p)

4.4 FunctionsautomorphicunderthesubgroupFo(p)

4.5 ConstructionoffunctionsbelongingtoFo(p)

4.6 Thebehavioroffpunderthegeneratorsofг

4.7 Thefunction(τ)=△(qτ)/△(τ)

4.8 Theunivalentfunctionφ(τ)

4.9 Invarianceofφ(τ)undertransformationsofг0(q)

4.1 0Thefunctionjpexpressedasapolynomialinφ

ExercisesforChapter4

Chapter5 Rademacher'sseriesforthepartitionfunction

5.1 Introduction

5.2 Theplanoftheproof

5.3 Dedekind'sfunctionalequationexpressedintermsofF

5.4 Fareyfractions

5.5 Fordcircles

5.6 Rademacher'spathofintegration

5.7 Rademacher'sconvergentseriesforp(n)

ExercisesforChapter5

Chapter6 Modularformswithmultiplicativecoefficients

6.1 Introduction

6.2 Modularformsofweightk

6.3 Theweightformulaforzerosofanentiremodularform

6.4 RepresentationofentireformsintermsofG4andG6

6.5 ThelinearspaceMkandthesubspaceMk.o

6.6 Classificationofentireformsintermsoftheirzeros

6.7 TheHeckeoperatorsTn

6.8 Transformationsofordern

6.9 BehaviorofTnfunderthemodulargroup

6.10 MultiplicativepropertyofHeckeoperators

6.11 EigenfunctionsofHeckeoperators

6.12 Propertiesofsimultaneouseigenforms

6.13 Examplesofnormalizedsimultaneouseigenforms

6.14 RemarksonexistenceofsimultaneouseigenformsinM2k.0

6.15 EstimatesfortheFouriercoefficientsofentireforms

6.16 ModularformsandDirichletseries

Exerci